Simultaneous inversion of induction data for dielectric permittivity and electric conductivity

ABSTRACT

A method of inverting induction logging data for evaluating the properties of underground formations surrounding a borehole, the data including induction voltage measurements obtained from a tool placed close to the formations of interest, the method includes: (a) defining a relationship relating the induction voltage to wave number, dielectric permittivity and conductivity; defining a cubic polynomial expansion of the relationship; and solving the cubic polynomial relationship using the voltage measurements to obtain values for conductivity that includes skin-effect correction, and apparent dielectric permittivity; and (b) using the obtained values for conductivity and apparent dielectric permittivity to derive a simulated value of induction voltage; determining the difference between the simulated value of the induction voltage and the measured induction voltage; and iteratively updating the values of conductivity and dielectric permittivity used for the derivation of the simulated value of induction voltage to minimize its difference with respect to the measured value.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a continuation-in-part of U.S. application Ser. No.12/404,454, filed Mar. 16, 2009, now U.S. Pat. No. 8,135,542, whichclaims the benefit under 35 U.S.C. 119 of European patent applicationNo. 08153574.2, filed Mar. 28, 2008, both applications incorporated byreference herewith in their entirety.

TECHNICAL FIELD

This invention relates to techniques for the inversion of inductionlogging data such as that used for evaluation of underground formationssurrounding a borehole. In particular, the invention relates to thesimultaneous inversion of induction data to derive the dielectricpermittivity and electric conductivity of the formation.

BACKGROUND ART

When drilling boreholes in the oil and gas industry, the nature of theunderground formations surrounding the borehole is typically obtained bymaking physical and/or chemical measurements with tools (often calledsondes) located in the borehole, the measurement responses of which areaffected by the properties of the formations. A series of suchmeasurements made along the length of the borehole is known as a log andone common form of log is that of a measurement relating to theelectrical resistivity of the formation. Resistivity logging techniquesare typically classified as either laterologs or induction logs.

Induction logs use an electric coil in the sonde to generate analternating current loop in the formation by induction. The alternatingcurrent loop, in turn, induces a current in a receiving coil locatedelsewhere on the sonde. The amount of current in the receiving coil isproportional to the intensity of current loop, hence to the conductivity(reciprocal of resistivity) of the formation. Multiple transmitting andreceiving coils can be used to focus formation current loops bothradially (depth of investigation) and axially (vertical resolution).Known types of induction logging sondes are the 6FF40 sonde which ismade up of six coils with a nominal spacing of 40 inches, and so-calledarray induction tools. These comprise a single transmitting coil and alarge number of receiving coils. Radial and axial focusing is performedby software rather than by the physical layout of coils.

Induction tools date back to the late 1940s and have so far been basedon an assumption of negligible dielectric effects in induction-tooldesign, processing and interpretation. Induction tools have become theindustry mainstay resistivity-saturation measurement since theirintroduction in the 1950s. The fundamental feature is a directmeasurement of the electric conductivity deep in the formation. Thebasic measurement is mostly unperturbed by any parasitic effects andtherefore is quite easily interpreted.

Recent developments of induction tools have provided accuratemeasurements of in-phase and quadrature signals. The quadrature signalis used to provide a skin-effect correction to the in-phase signal.Traditionally, induction-tool processing and interpretation neglectsdielectric effects completely. The present invention re-introducesdielectric effects into induction-tool processing and proposes twosimple inversion algorithms that can be used to determine a dielectricpermittivity and electric conductivity from the in-phase and quadraturesignals simultaneously.

SUMMARY OF THE INVENTION

A first aspect of the invention provides a method of inverting inductionlogging data for evaluating the properties of underground formationssurrounding a borehole, the data comprising induction voltagemeasurements obtained from a tool placed close to the formations ofinterest, the method comprising:

(a) defining a relationship relating the induction voltage to wavenumber, dielectric permittivity and conductivity;

-   -   defining a cubic polynomial expansion of the relationship; and    -   solving the cubic polynomial relationship using the voltage        measurements to obtain values for conductivity that includes        skin-effect correction, and apparent dielectric permittivity;        and        (b) using the obtained values for conductivity and apparent        dielectric permittivity to derive a simulated value of induction        voltage;    -   determining the difference between the simulated value of the        induction voltage and the measured induction voltage; and    -   iteratively updating the values of conductivity and dielectric        permittivity used for the derivation of the simulated value of        induction voltage to minimise its difference with respect to the        measured value.

Preferably, the relationship relating induction voltage to wave number,dielectric permittivity and conductivity is:

$k = {\frac{\omega}{c}\sqrt{\mu_{r}}\sqrt{ɛ_{r} + {{\mathbb{i}}\frac{\sigma}{{\omega ɛ}_{0}}}}}$where k is wave number, ω is circular frequency, c is the speed of lightin a vacuum, μ_(r) is relative magnetic permeability, ∈_(r) is relativepermittivity, ∈₀ is the absolute dielectric permittivity of a vacuum andσ is conductivity.

The method typically comprises deriving the roots of the cubicpolynomial and using at least one of the roots to obtain the wavenumber. Roots giving physically impossible values for parameters ofinterest can be ignored.

The step of iteratively updating the values of conductivity anddielectric permittivity used for the derivation of the simulated valueof induction voltage to minimise its difference with respect to themeasured value preferably comprises minimising the squared differencebetween measured induction voltage U_(meas) and their simulatedreproduction U_(simul) according to the relationship:

$L = {\left. {\frac{1}{2}\left( {U_{meas} - U_{simul}} \right)^{2}}\Rightarrow{\frac{\mathbb{d}}{\mathbb{d}k}L} \right. = {{\left( {U_{meas} - U_{simul}} \right)\frac{\mathbb{d}U_{simul}}{\mathbb{d}k}} = 0.}}$

In some embodiments, the iteration can use a Newton-Raphson algorithm

${U_{meas} - U_{simul}^{({n - 1})}} = {\frac{\mathbb{d}U_{simul}^{({n - 1})}}{\mathbb{d}k}\left( {k^{(n)} - k^{({n - 1})}} \right)}$

The iterative update for the wave number k^((n)) is derived from therelationship:k ^((n)) =k ^((n-1))+(U _(meas) −U _(simul) ^((n-1)))/(dU _(simul)^((n-1)) /dk).

In some embodiments, the iteration can use a quadratic algorithm:

$U_{meas} = {U_{simul}^{({n - 1})} + {\frac{\mathbb{d}U_{simul}^{({n - 1})}}{\mathbb{d}k}\left( {k^{(n)} - k^{({n - 1})}} \right)} + {\frac{1}{2}\frac{\mathbb{d}^{2}U_{simul}^{({n - 1})}}{\mathbb{d}k^{2}}\left( {k^{(n)} - k^{({n - 1})}} \right)^{2}}}$and the iterative update for the wave number k^((n)) is derived from therelationship:

$\begin{matrix}{k^{(n)} = {k^{({n - 1})} - \frac{{\mathbb{d}U_{simul}^{({n - 1})}}/{\mathbb{d}k}}{{\mathbb{d}^{2}U_{simul}^{({n - 1})}}/{\mathbb{d}k^{2}}}}} \\{\pm \sqrt{\left( \frac{{\mathbb{d}U_{simul}^{({n - 1})}}/{\mathbb{d}k}}{{\mathbb{d}^{2}U_{simul}^{({n - 1})}}/{\mathbb{d}k^{2}}} \right)^{2} - {2\frac{U_{simul}^{({n - 1})} - U_{meas}}{{\mathbb{d}^{2}U_{simul}^{({n - 1})}}/{\mathbb{d}k^{2}}}}}}\end{matrix}$

In some embodiments, the iteration can use a cubic algorithm:

$U_{meas} = {U_{simul}^{({n - 1})} + {\frac{\mathbb{d}U_{simul}^{({n - 1})}}{\mathbb{d}k}\left( {k^{(n)} - k^{({n - 1})}} \right)} + {\frac{1}{2}\frac{\mathbb{d}^{2}U_{simul}^{({n - 1})}}{\mathbb{d}k^{2}}\left( {k^{(n)} - k^{({n - 1})}} \right)^{2}} + {\frac{1}{6}\frac{\mathbb{d}^{3}U_{simul}^{({n - 1})}}{\mathbb{d}k^{3}}\left( {k^{(n)} - k^{({n - 1})}} \right)^{3}}}$and the iterative update for the wave number k^((n)) is derived from therelationship:

${\left( {k^{(n)} - k^{({n - 1})}} \right)^{3} + {3\frac{{\mathbb{d}^{2}U_{simul}^{({n - 1})}}/{\mathbb{d}k^{2}}}{{\mathbb{d}^{3}U_{simul}^{({n - 1})}}/{\mathbb{d}k^{3}}}\left( {k^{(n)} - k^{({n - 1})}} \right)^{2}} + {6\frac{{\mathbb{d}U_{simul}^{({n - 1})}}/{\mathbb{d}k}}{{\mathbb{d}^{3}U_{simul}^{({n - 1})}}/{\mathbb{d}k^{3}}}\left( {k^{(n)} - k^{({n - 1})}} \right)} + {6\frac{U_{simul}^{({n - 1})} - U_{meas}}{{\mathbb{d}^{3}U_{simul}^{({n - 1})}}/{\mathbb{d}k^{3}}}}} = 0$

Furthermore, higher-order polynomials may be used in an analogousmanner. Alternately, other, non-polynomial approximations may be used.

Preferably, the voltage measurements comprise in-phase and quadraturemeasurements of substantially the same accuracy and resolution, whichcan be processed in parallel.

Where the induction logging data is obtained from a three-coil toolcomprising a transmitter coil with magnetic dipole moment M_(T), a mainreceiver coil with magnetic dipole moment M₁ at distance r₁ from thetransmitter coil, and a bucking coil with a magnetic moment M₂ atdistance r₂ from the transmitter coil, the dipole moments being alignedparallel to the axis of the tool, the cubic polynomial expansion cancomprise:

$U_{l} = {{- {\mathbb{i}\omega\mu}}\;\frac{M_{T}}{2\pi}\left( {{\left( {\frac{M_{1}}{2r_{1}} - \frac{M_{2}}{2r_{2}}} \right)k^{2}} + {{{\mathbb{i}}\left( {\frac{M_{1}}{3} - \frac{M_{2}}{3}} \right)}k^{3}}} \right)}$wherein U₁ is the induction voltage.

In this case, the simulated value of induction voltage can be derivedaccording to the relationship

$U_{l} = {{- {\mathbb{i}\omega\mu}}\;\frac{M_{T}}{2\pi}\left( {{\frac{M_{1}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{1}}}{r_{1}^{3}}\left( {1 - {{\mathbb{i}}\;{kr}_{1}}} \right)} - {\frac{M_{2}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{2}}}{r_{2}^{3}}\left( {1 - {{\mathbb{i}}\;{kr}_{2}}} \right)}} \right)}$using a sensitivity function derived according to the relationship:

$\frac{\mathbb{d}U_{l}}{\mathbb{d}k} = {{- {\mathbb{i}\omega\mu}}\;\frac{M_{T}}{2\pi}\left( {\frac{M_{1}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{1}}}{r_{1}} - \frac{M_{2}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{2}}}{r_{2}}} \right)k}$where e^(ikr) is the full electromagnetic wave.

Where the dipole moments are aligned transverse to the axis of the tool,the cubic polynomial expansion can comprise:

$U_{t} = {{- {\mathbb{i}\omega\mu}}\;\frac{M_{T}}{4\pi}\left( {{\left( {\frac{M_{1}}{2r_{1}} - \frac{M_{2}}{2r_{2}}} \right)k^{2}} + {2{{\mathbb{i}}\left( {\frac{M_{1}}{3} - \frac{M_{2}}{3}} \right)}k^{3}}} \right)}$wherein U_(t) is the induction voltage.

In this case, the simulated value of induction voltage can be derivedaccording to the relationship

$U_{t} = {{- {\mathbb{i}\omega\mu}}\;\frac{M_{T}}{4\pi}\left( {{\frac{M_{1}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{1}}}{r_{1}^{3}}\left( {1 - {{\mathbb{i}}\;{kr}_{1}} - {k^{2}r_{1}^{2}}} \right)} - {\frac{M_{2}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{2}}}{r_{2}^{3}}\left( {1 - {{\mathbb{i}}\;{kr}_{2}} - {k^{2}r_{2}^{2}}} \right)}} \right)}$using a sensitivity function derived according to the relationship:

$\frac{\mathbb{d}U_{t}}{\mathbb{d}k} = {{\mathbb{i}\omega\mu}\;\frac{M_{T}}{4\pi}\left( {{\frac{M_{1}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{1}}}{r_{1}}\left( {1 + {{\mathbb{i}}\;{kr}_{1}}} \right)} - {\frac{M_{2}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{2}}}{r_{2}}\left( {1 + {{\mathbb{i}}\;{kr}_{2}}} \right)}} \right){k.}}$

Further aspects of the invention, for example, the acquisition of data,will be apparent from the following description.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a schematic diagram of one embodiment of a three-coilinduction sonde in a borehole;

FIG. 2 shows a schematic diagram of another embodiment of a three-coilinduction sonde in a borehole: and

FIG. 3 shows schematically the quadratic and cubic terms considered inknown induction processing with skin-effect correction.

MODE(S) FOR CARRYING OUT THE INVENTION

FIG. 1 shows a schematic diagram of one embodiment of a three-coilinduction sonde in a borehole. The sonde 10 comprises a transmitter coil12 and a main receiver coil 14 spaced from the transmitter 12 on thesonde 10. A bucking coil 16 is positioned on the sonde 10 between thetransmitter 12 and receiver 14. In this embodiment, the dipole momentsof the coils 12, 14, 16 are parallel to the sonde axis (longitudinallypolarised). The transmitter and receiver coils 12, 14 are arranged so asto have the same polarisation whereas the bucking sonde 16 has apolarisation that is reversed. An example of such a sonde is the ArrayInduction Tool (AIT) of Schlumberger.

FIG. 2 shows a schematic diagram of another embodiment of a three-coilinduction sonde 20. In this embodiment, the dipole moments of the coils22, 24, 26 are perpendicular to the sonde axis (transversely polarised).Again, the bucking sonde 26 has a polarisation that is reversed comparedto the transmitter 22 and receiver 24. An example of such a sonde is theRtScanner of Schlumberger and the TriDEX tool of Baker-Hughes.

Operation of the tools shown in FIGS. 1 and 2 to obtain inductionvoltage data is well-known. FIG. 3 shows schematically the quadratic andcubic terms considered in known induction processing with skin-effectcorrection.

An induction tool generates an electric field in the surroundingformation by an alternating current in a transmitter coil, whichconstitutes an oscillating magnetic dipole. The magnetic dipolegenerates a magnetic field, which is described by the Maxwell-Ampereequation:{right arrow over (∇)}×{right arrow over (H)}=∂ _(t) {right arrow over(D)}+{right arrow over (j)}  (1)

This equation combines the displacement current {right arrow over (D)}with the galvanic-current density {right arrow over (j)}. Thedisplacement current is related to the electric field {right arrow over(E)} by the dielectric permittivity ∈, while Ohm's law in microscopicform relates the galvanic-current density {right arrow over (j)} and theelectric field {right arrow over (E)} through the electric conductivityσ. Only the transmitter current is explicitly carried through assource-current density {right arrow over (j)}₀. The induction sondeserves to provide the electric conductivity for saturation estimation.

J. D. Jackson, Classical Electrodynamics, Wiley 3^(rd) Ed. (1998), ch.9, eq. (9.35), p. 413. derives the close-form solution of a magneticpoint-dipole radiator in a homogeneous, isotropic medium. The magneticfield of this dipole is:

$\begin{matrix}{{\overset{->}{H}\left( \overset{->}{r} \right)} = {\frac{{\mathbb{e}}^{{\mathbb{i}}\;{kr}}}{4\pi\; r^{3}}\begin{pmatrix}{{\hat{n} \times \left( {\overset{->}{M} \times \hat{n}} \right)k^{2}r^{2}} +} \\{\left( {{3{\hat{n}\left( {\overset{->}{M} \cdot \hat{n}} \right)}} - \overset{->}{M}} \right)\left( {1 - {{\mathbb{i}}\;{kr}}} \right)}\end{pmatrix}}} & (2)\end{matrix}$with the magnetic-dipole moment {right arrow over (M)} and the distancevector between radiator and observation point {right arrow over (r)}with normal direction {circumflex over (n)}={right arrow over (r)}/r.The wave number is given in terms of the circular frequency ω with theuniversal time dependence e^(−iωt) and the electromagnetic materialparameters ∈, σ, μ:

$\begin{matrix}{k = {\sqrt{{\omega\mu}\left( {{\omega ɛ} + {\mathbb{i}\sigma}} \right)} = {\frac{\omega}{c}\sqrt{\mu_{r}}\sqrt{ɛ_{r} + {{\mathbb{i}}\frac{\sigma}{{\omega ɛ}_{0}}}}}}} & (3)\end{matrix}$where c is the speed of light in vacuum. Usually, the relative magneticpermeability of the formation is simply set to μ_(r)=1.

The magnetic field from a transmitter dipole {right arrow over (M)}_(T)will induce a voltage U in a receiver magnetic dipole {right arrow over(M)}_(R) according to Faraday's law:U=∂ _(t) {right arrow over (B)}·{right arrow over (M)} _(R) =−iωμ{rightarrow over (H)}·{right arrow over (M)} _(R)  (4)

The magnetic field from equation (2) provides the close-form solutionfor the induced voltage, which is symmetric in transmitter and receiverdipole moments:

$\begin{matrix}{U = {{- {\mathbb{i}\omega\mu}}\frac{{\mathbb{e}}^{{\mathbb{i}}\;{kr}}}{4\pi\; r^{3}}\begin{pmatrix}{{{\left( {{\overset{->}{M}}_{T} \times \hat{n}} \right) \cdot \left( {{\overset{->}{M}}_{R} \times \hat{n}} \right)}k^{2}r^{2}} +} \\{\left( {{3\left( {{\overset{->}{M}}_{T} \cdot \hat{n}} \right)\left( {{\overset{->}{M}}_{R} \cdot \hat{n}} \right)} - {{\overset{->}{M}}_{T} \cdot {\overset{->}{M}}_{R}}} \right)\left( {1 - {{\mathbb{i}}\;{kr}}} \right)}\end{pmatrix}}} & (5)\end{matrix}$

This solution is used to describe and invert the three-coil point-dipoleinduction sondes for either polarization.

A longitudinally polarized induction sonde, such as the AIT (see FIG. 1)has dipole moments parallel to the tool axis: {right arrow over(M)}_(T)∥{right arrow over (M)}_(R)∥{right arrow over (r)}; hence({right arrow over (M)}_(T)×{circumflex over (n)})=0=({right arrow over(M)}_(R)×{circumflex over (n)}) and ({right arrow over(M)}_(T)×n{circumflex over (n)})·({right arrow over (M)}_(R)·{circumflexover (n)})={right arrow over (M)}_(T)·{right arrow over (M)}_(R). On theother hand, the RtScanner sonde (see FIG. 2) has transversely polarizeddipoles with {right arrow over (M)}_(T)∥{right arrow over(M)}_(R)⊥{right arrow over (r)}; hence ({right arrow over(M)}_(T)·n{circumflex over (n)})=0=({right arrow over(M)}_(R)·{circumflex over (n)}) and ({right arrow over(M)}_(T)×{circumflex over (n)})·({right arrow over (M)}_(R)×{circumflexover (n)})={right arrow over (M)}_(R)×{right arrow over (M)}_(T)·{rightarrow over (M)}_(R). These vectorial projections simplify theinduced-voltage expression (5) for either polarization (l or t):

$\begin{matrix}{U_{l} = {{- {\mathbb{i}\omega\mu}}\frac{{\mathbb{e}}^{{\mathbb{i}}\;{kr}}}{4\pi\; r^{3}}2{{\overset{->}{M}}_{T} \cdot {{\overset{->}{M}}_{R}\left( {1 - {{\mathbb{i}}\;{kr}}} \right)}}}} & \left( {6l} \right) \\{U_{t} = {{\mathbb{i}\omega\mu}\frac{{\mathbb{e}}^{{\mathbb{i}}\;{kr}}}{4\pi\; r^{3}}{{\overset{->}{M}}_{T} \cdot {{\overset{->}{M}}_{R}\left( {1 - {{\mathbb{i}}\;{kr}} - {k^{2}r^{2}}} \right)}}}} & \left( {6t} \right)\end{matrix}$

These two cases can be considered in parallel, yet independent from eachother.

In the low-frequency or long-wavelength limit kr<<1, the exponential isexpanded in a rapidly converging Taylor series. This series is to cubicorder:

$\begin{matrix}{{\mathbb{e}}^{{\mathbb{i}}\;{kr}} \cong {1 + {{\mathbb{i}}\;{kr}} - \frac{k^{2}r^{2}}{2} - {{\mathbb{i}}\;\frac{k^{3}r^{3}}{6}} + \ldots}} & (7)\end{matrix}$

This Taylor expansion is multiplied with the corresponding parenthesesin equations (6) to yield the two cubic polynomials:

$\begin{matrix}{{\left( {1 + {{\mathbb{i}}\;{kr}} - \frac{k^{2}r^{2}}{2} - {{\mathbb{i}}\;\frac{k^{3}r^{3}}{6}}} \right)\left( {1 - {{\mathbb{i}}\;{kr}}} \right)} \cong {1 + \frac{k^{2}r^{2}}{2} + {{\mathbb{i}}\frac{k^{3}r^{3}}{3}}}} & \left( {8l} \right) \\{{\left( {1 + {{\mathbb{i}}\;{kr}} - \frac{k^{2}r^{2}}{2} - {{\mathbb{i}}\;\frac{k^{3}r^{3}}{6}}} \right)\left( {1 - {{\mathbb{i}}\;{kr}} - {k^{2}r^{2}}} \right)} \cong {1 - \frac{k^{2}r^{2}}{2} - {2{\mathbb{i}}\;\frac{k^{3}r^{3}}{3}}}} & \left( {8t} \right)\end{matrix}$

The leading term 1 dominates these expansions; however, it does notcontain any formation information. The next leading term is k²r², whichis proportional to the complex-valued dielectric constant, or withnegligible permittivity proportional to the electric conductivity.Therefore, the leading term must be eliminated.

The leading term is the same in both polarizations. This term must beeliminated to render the measurement sensitive to the formationconductivity. This elimination is accomplished by a bucking receiver,leading to the three-coil design. This three-coil design is the basicbuilding block of all modern array induction tools. The main and buckingreceiver are at distances r₁ and r₂ with magnetic dipole moments M₁ andM₂. The combined signal voltage is then:

$\begin{matrix}{U_{l} = {{- {\mathbb{i}\omega\mu}}\;{\frac{M_{T}}{2\pi}\left\lbrack {{\frac{M_{1}}{r_{1}^{3}}\left( {1 + \frac{k^{2}r_{1}^{2}}{2} + {{\mathbb{i}}\;\frac{k^{3}r_{1}^{3}}{3}}} \right)} - {\frac{M_{2}}{r_{2}^{3}}\left( {1 + \frac{k^{2}r_{2}^{2}}{2} + {{\mathbb{i}}\;\frac{k^{3}r_{2}^{3}}{3}}} \right)}} \right\rbrack}}} & \left( {9l} \right) \\{U_{t} = {{\mathbb{i}\omega\mu}{\frac{M_{T}}{4\pi}\left\lbrack {{\frac{M_{1}}{r_{1}^{3}}\left( {1 - \frac{k^{2}r_{1}^{2}}{2} - {2{\mathbb{i}}\frac{k^{3}r_{1}^{3}}{3}}} \right)} - {\frac{M_{2}}{r_{2}^{3}}\left( {1 - \frac{k^{2}r_{2}^{2}}{2} - {2{\mathbb{i}}\frac{k^{3}r_{2}^{3}}{3}}} \right)}} \right\rbrack}}} & \left( {9t} \right)\end{matrix}$

The condition eliminating the leading term in either equation is:

$\begin{matrix}{{\frac{M_{1}}{r_{1}^{3}} - \frac{M_{2}}{r_{2}^{3}}} = 0} & (10)\end{matrix}$which determines the bucking-receiver dipole moment in terms of the mainreceiver moment and the distances to M₂=M₁r₂ ³/r₁ ³ or thebucking-receiver distance in terms of the main receiver distance and thetwo receiver dipole moments to r₂=r₁ ³√{square root over (M₂/M₁)}.

Usually, the main-receiver distance and dipole moment are fixed. Thebucking-receiver distance is then approximately fixed and its dipolemoment is determined in terms of number of turns of the two receivercoils. These number of turns are integer valued; so after fixing thedipole moments the bucking-receiver distance is adjusted to fulfil thebucking condition (10).

In the following analysis the two receivers will be carried through asindependent variables; the bucking condition (10) is implicitly assumedthroughout the rest of this study. The leading term is simply omittedfrom here on.

The dielectric permittivity ∈=∈_(r)∈₀ contains the relative permittivity∈_(r) without units and the absolute dielectric permittivity of vacuum∈₀(≅8.8542*10⁻¹² As/Vm). This absolute dielectric permittivity is a verysmall number, which is combined with the circular frequency ω(=2πf) intoa conductivity scale σ₀=ω∈₀. An induction frequency of 25 kHz leads to aconductivity scale of σ₀=ω∈₀≅1.4 μS/m. This conductivity scale is sosmall compared to typical formation conductivities thatdielectric-permittivity effects commonly are neglected in inductionprocessing and interpretation. Hence the Maxwell-Ampere equation (1) issimplified to:{right arrow over (∇)}×{right arrow over (H)}≅σ{right arrow over(E)}+{right arrow over (j)} ₀  (11)

This induction approximation simplifies the wave equation and hence thewave number:

$\begin{matrix}{k = {{\frac{\omega}{c}\sqrt{\mu_{r}}\sqrt{ɛ_{r} + {{\mathbb{i}}\frac{\sigma}{{\omega ɛ}_{0}}}}} \cong \sqrt{{\mathbb{i}\omega\mu}_{0}\sigma}}} & (12)\end{matrix}$using the identity c²=1/∈₀μ₀ from the wave equation. The relativemagnetic permeability has been set to μ_(r)=1.

In this approximation the squared wave number is directly proportionalto the electric conductivity. The induced-voltage signals (9) become:

$\begin{matrix}\begin{matrix}{U_{l} = {{- {\mathbb{i}\omega\mu}}\;\frac{M_{T}}{2\pi}\left( {\frac{M_{1}}{2r_{1}} - \frac{M_{2}}{2r_{2}} + {{\mathbb{i}}\frac{k}{3}\left( {M_{1} - M_{2}} \right)}} \right)k^{2}}} \\{= {\omega^{2}\mu^{2}\frac{M_{T}}{2\pi}\left( {\frac{M_{1}}{2r_{1}} - \frac{M_{2}}{2r_{2}} + {{\mathbb{i}}\frac{k}{3}\left( {M_{1} - M_{2}} \right)}} \right)\sigma}}\end{matrix} & \left( {13l} \right) \\\begin{matrix}{U_{t} = {{- {\mathbb{i}\omega\mu}}\;\frac{M_{T}}{4\pi}\left( {\frac{M_{1}}{2r_{1}} - \frac{M_{2}}{2r_{2}} + {2{\mathbb{i}}\frac{k}{3}\left( {M_{1} - M_{2}} \right)}} \right)k^{2}}} \\{= {\omega^{2}\mu^{2}\frac{M_{T}}{4\pi}\left( {\frac{M_{1}}{2r_{1}} - \frac{M_{2}}{2r_{2}} + {2{\mathbb{i}}\frac{k}{3}\left( {M_{1} - M_{2}} \right)}} \right)\sigma}}\end{matrix} & \left( {13t} \right)\end{matrix}$

To lowest order in the wave number the apparent conductivity σ_(a) isthus given by the measured in-phase or R-signal voltage for eitherpolarization:

$\begin{matrix}{\sigma_{al} = {\frac{2\pi}{\omega^{2}\mu^{2}M_{T}}\frac{1}{{{M_{1}/2}r_{1}} - {{M_{2}/2}r_{2}}}U_{l}}} & \left( {14l} \right) \\{\sigma_{at} = {\frac{4\pi}{\omega^{2}\mu^{2}M_{T}}\frac{1}{{{M_{1}/2}r_{1}} - {{M_{2}/2}r_{2}}}U_{t}}} & \left( {14t} \right)\end{matrix}$

The next-order term contains an additional wave-number factor.

The real and imaginary part of this induced-voltage signal constitutesthe in-phase and quadrature signals. The cubic term in equations (13)contains an imaginary unit in the square root, which leads to equal realand imaginary parts. The real part appears negative and thus issubtracted from the leading, quadratic-order term. This subtractionartificially reduces the apparent conductivity or increases the apparentresistivity.

$\begin{matrix}{{k \cong \sqrt{{\mathbb{i}\omega\mu}_{0}\sigma}} = {\left( {1 + i} \right)\sqrt{\frac{{\omega\mu}_{0}\sigma}{2}}}} & (15)\end{matrix}$

Hence the real and imaginary part, or the R- and the X-signalcontributions from this term are equal. If higher-order terms are trulynegligible the X-signal term is independent. The contribution of thereal, in-phase part is superposed on the leading term, causing adistortion that is attributed to the skin effect. By itself, thisdistortion cannot be filtered from the in-phase signal. However, thequadrature signal constitutes exactly the same signal as the distortionand this can be used to correct the apparent conductivity. (FIG. 3)

The imaginary part can be directly and independently measured asquadrature signal. As part of advanced induction processing it can beadded to the leading in-phase term, where it will effectively compensatefor the skin-effect distortion. This correction is used in modernarray-induction tools, providing the skin-effect-corrected apparentconductivity. This correction gives good results for conductivitiesbelow 100 mS/m (resistivities above 10 Ωm). For more conductiveformations the skin-effect correction rapidly becomes larger, so thataround 1000 mS/m (resistivity of 1 Ωm) it is around 25% for a 27″-39″receiver-coil pair. In this situation higher-order terms of the wavenumber are no longer negligibly small.

The quadrature signal can provide an accurate correction for the cubicin-phase distortion. For such a correction it is important that bothin-phase and quadrature signal be measured with equally high accuracyand precision since it is impossible to infer any such correction froman auxiliary measurement at a different frequency. Notably, theformation properties usually are slightly frequency dependent, so that ahigh-precision correction cannot reliably be derived from adifferent-frequency measurement.

The known induction tools are designed with the assumption thatdielectric effects are negligibly small at the comparatively lowoperating frequencies. This assumption has reduced the complex-valuedcharacter for the formation measurement to a simpler, real-valuedconductivity measurement. The complex-valued measurement is used toimprove the apparent conductivity by the skin-effect correction.However, recent processing and interpretation of some strange inductionlogs using the known techniques have led to observations of unexpectedlylarge dielectric permittivities in some shales with values between 10000and 50000 for measurements at 26 kHz and at 52 kHz. Theinduction-processing according to the invention explicitly includespermittivity and integrates it at the most fundamental level with theskin-effect correction.

The wave number for Maxwell's equations includes both dielectricpermittivity and electric conductivity as real and imaginary part of acomplex-valued quantity. Only the low-frequency limit considers thepermittivity negligible and reduces the wave number to a complex-valuedfunction of conductivity only. In this limit the square of the wavenumber is purely imaginary and directly proportional to the electricconductivity.

$\begin{matrix}{k^{2} = {{\frac{\omega^{2}}{c^{2}}{\mu_{r}\left( {ɛ_{r} + {{\mathbb{i}}\frac{\sigma}{{\omega ɛ}_{0}}}} \right)}} \cong {{\mathbb{i}}\frac{\omega^{2}}{c^{2}}\mu_{r}\frac{\sigma}{{\omega ɛ}_{0}}}}} & \left( 12^{\prime} \right)\end{matrix}$

The three-coil induction sonde measures the combined voltage between thereceivers, as given in equation (3) and low-frequency, cubic polynomialapproximated in equation (6). This induced voltage is given in terms ofthe wave number; using the complete expression from equation (7)includes the full permittivity dependence.

The skin-effect correction uses the cubic term in the wave number andthe fact that the imaginary and real parts were equal due to the squareroot of the imaginary unit:

$\begin{matrix}{\sqrt{\mathbb{i}} = {\pm \frac{1 + {\mathbb{i}}}{\sqrt{2}}}} & (8)\end{matrix}$

With the dielectric permittivity included in the square root, thisassumption of equal real and imaginary parts does not hold any more.However, the cubic polynomial expansion (9) of the voltage in terms ofthe wave number still holds.

$\begin{matrix}{U_{l} = {{- {\mathbb{i}\omega\mu}}\;\frac{M_{T}}{2\pi}\left( {{\left( {\frac{M_{1}}{2r_{1}} - \frac{M_{2}}{2r_{2}}} \right)k^{2}} + {{{\mathbb{i}}\left( {\frac{M_{1}}{3} - \frac{M_{2}}{3}} \right)}k^{3}}} \right)}} & \left( {9l^{\prime}} \right) \\{U_{t} = {{- {\mathbb{i}\omega\mu}}\;\frac{M_{T}}{4\pi}\left( {{\left( {\frac{M_{1}}{2r_{1}} - \frac{M_{2}}{2r_{2}}} \right)k^{2}} + {2{{\mathbb{i}}\left( {\frac{M_{1}}{3} - \frac{M_{2}}{3}} \right)}k^{3}}} \right)}} & \left( {9t^{\prime}} \right)\end{matrix}$

The cubic polynomial can be solved in closed form as is described in M.Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions”, Dover(1964, 9^(th) Ed. 1972), eq. 3.8.2, p. 17, i.e., the roots of thepolynomial are given as simple arithmetic expressions. Considering thatthe cubic polynomial coefficients are already complex valued theinversion for the full wave number including the dielectric permittivitydoes not generate any further computational overhead. For the cubicpolynomial inversion the full wave number (3) is considered, albeit withthe assumption for the relative magnetic permeability μ_(r)=1.

A general cubic polynomial equation has the form:a ₃ z ³ +a ₂ z ² +a ₁ z+a ₀ =b  (16)with complex-valued coefficients. This polynomial is simplified to:

$\begin{matrix}{{z^{3} + {\frac{a_{2}}{a_{3}}z^{2}} + {\frac{a_{1}}{a_{3}}z} + \frac{a_{0} - b}{a_{3}}} = {{z^{3} + {c_{2}z^{2}} + {c_{1}z} + c_{0}} = 0}} & (17)\end{matrix}$

The new coefficients are used to define two auxiliary quantities:

$\begin{matrix}{{d = {\frac{c_{1}}{3} - \frac{c_{2}^{2}}{9}}}{e = {{\frac{1}{6}\left( {{c_{1}c_{2}} - {3c_{0}}} \right)} - \frac{c_{2}^{3}}{27}}}} & (18)\end{matrix}$

These two auxiliary quantities in turn define two additional quantities:

$\begin{matrix}{{f_{1} = \sqrt[3]{e + \sqrt{d^{3} + e^{2}}}}{f_{2} = \sqrt[3]{e - \sqrt{d^{3} + e^{2}}}}} & (19)\end{matrix}$

Finally the coefficients and the auxiliary quantities give the threeroots of the cubic polynomial:

$\begin{matrix}{z_{1} = {\left( {f_{1} + f_{2}} \right) - \frac{c_{2}}{3}}} & \left( {20a} \right) \\{z_{2} = {{{- \frac{1}{2}}\left( {f_{1} + f_{2}} \right)} - \frac{c_{2}}{3} + {{\mathbb{i}}\frac{\sqrt{3}}{2}\left( {f_{1} - f_{2}} \right)}}} & \left( {20b} \right) \\{z_{3} = {{{- \frac{1}{2}}\left( {f_{1} + f_{2}} \right)} - \frac{c_{2}}{3} - {{\mathbb{i}}\frac{\sqrt{3}}{2}\left( {f_{1} - f_{2}} \right)}}} & \left( {20c} \right)\end{matrix}$

Testing of this algorithm on the cubic polynomials for theinduced-voltage signals shows that only the first root (20a) providesthe physically correct solution for the wave number. The other rootslead to negative electric conductivities and so may be ignored.

This cubic polynomial solution implicitly accounts for skin effect andpermittivity. It no longer makes sense to distinguish in-phase andquadrature signals, since the permittivity and conductivity have beencombined to a complex-valued quantity that already is skewed in phaseagainst the measured signals. At the same time, the cubic polynomialinversion implicitly performs the act that was explicitly used tocorrect the quadratic-order apparent conductivity for the skin-effectdistortion.

Testing on synthetic data shows a systematic discrepancy in the apparentdielectric permittivity between the known input data and the invertedresults in certain circumstances: the inverted permittivities aresystematically higher than the input values; this discrepancy increasesrapidly at higher conductivities (lower resistivities). Around 10 Ωm thediscrepancy is about 550 while at 1 Ωm it becomes about 55000. Suchdiscrepancy may unacceptable for more conductive shales.

In this situation an iterative inversion of the full equations (6) forthe wave number can provide accurate and reliable results for bothapparent electric conductivity and dielectric permittivity. The cubicpolynomial solution serves as initial estimate.

Formally, the measured signals consist of two independent, real-valuedsignals that are inverted for two independent, real valued quantities:dielectric permittivity and electric conductivity. Therefore, naivelythe inversion constitutes a two-dimensional, real-valued problem withfour independent sensitivity derivatives. However, the target quantitiesconstitute real and imaginary part of a complex-valued parameter. Themeasured-voltage signal is a complex-valued function of this parameter;in fact, it is a holomorphic function that obeys the Cauchy-Riemannidentities.

In the theory of complex functions (see, for example, L. Ahlfors,“Complex Analysis”, 2^(nd) Edition, McGraw Hill (1966), Ch. 2.1, pp.21-33), elementary functions are a special class of holomorphicfunctions. Elementary functions are defined as finite-order sums orproducts of polynomials, exponentials and logarithms. Hencetrigonometric and hyperbolic functions and their inverse all areelementary functions. These functions may have complex-valued argumentsand then become complex-valued.

A complex-valued function w(=u+iv) of a complex argument z(=x+iy) iscalled a holomorphic function if it depends only on the complex variablez, but not its complex conjugate z(=x−iy). In the case of theelectromagnetic Green function the holomorphism is important toillustrate that the electromagnetic signal depends only on the complexdielectric constant with a positive imaginary part. The positiveimaginary part describes the dissipation of the electromagnetic signalin lossy media; a negative imaginary part would describe an exponentialgrowth of the signal in blatant violation of energy conservation.

A holomorphic function w(z)=u(x,y)+iv(x,y) is differentiable and obeysthe Cauchy-Riemann identities:

$\begin{matrix}{{\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}}{\frac{\partial u}{\partial y} = {- \frac{\partial v}{\partial x}}}} & (I)\end{matrix}$

These identities reflect the fact that the holomorphic function does notdepend on z(=x−iy).

These holomorphic functions and their Cauchy-Riemann identities arewell-established general mathematical concepts in the theory of complexfunctions, but to date their manifestation in the physics ofelectromagnetic induction measurements has not been appreciated. Theelectromagnetic Green function describes the complex-valuedtransmitter-receiver signals as holomorphic function of the complexdielectric constant. Hence the in-phase (R−) and quadrature (X−)signals,U_(R) and U_(X), as real and imaginary part of the received inductionsignal obey Cauchy-Riemann identities in their dependence on theelectric conductivity and dielectric permittivity:

$\begin{matrix}{{\frac{\partial U_{R}}{\partial\sigma} = \frac{\partial U_{X}}{{\omega ɛ}_{0}{\partial ɛ_{r}}}}{\frac{\partial U_{X}}{\partial\sigma} = {- \frac{\partial U_{R}}{{\omega ɛ}_{0}{\partial ɛ_{r}}}}}} & ({II})\end{matrix}$

The in-phase and quadrature measurements, U_(R) and U_(X), can besimultaneously inverted for the relative dielectric permittivity and theelectric conductivity. As alternative to the traditionalconductivity-conversion method or the cubic polynomial inversion, we mayuse an iterative inversion algorithm. Such an iterative inversionalgorithm provides considerable advantages: it is numerically morestable, converges rapidly and involves the explicit Green-functionexpressions.

The holomorphic character eliminates two of the four sensitivityderivatives and combines the remaining two sensitivities into a single,complex-valued derivative. The close-form solution for theinduced-voltage signals readily provides close-form expressions for thesensitivity to the wave number:

$\begin{matrix}{\mspace{79mu}{U_{l} = {{- {\mathbb{i}\omega\mu}}\;\frac{M_{T}}{2\pi}\left( {{\frac{M_{1}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{1}}}{r_{1}^{3}}\left( {1 - {{\mathbb{i}}\;{kr}_{1}}} \right)} - {\frac{M_{2}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{2}}}{r_{2}^{3}}\left( {1 - {{\mathbb{i}}\;{kr}_{2}}} \right)}} \right)}}} & \left( {21\; l} \right) \\{U_{t} = {{- {\mathbb{i}}}\;{\omega\mu}\;\frac{M_{T}}{4\pi}\left( {{\frac{M_{1}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{1}}}{r_{1}^{3}}\left( {1 - {{\mathbb{i}}\;{kr}_{1}} - {k^{2}r_{1}^{2}}} \right)} - {\frac{M_{2}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{2}}}{r_{2}^{3}}\left( {1 - {{\mathbb{i}}\;{kr}_{2}} - {k^{2}r_{2}^{2}}} \right)}} \right)}} & \left( {21\; t} \right) \\{\mspace{79mu}{\frac{\mathbb{d}U_{l}}{\mathbb{d}k} = {{- {\mathbb{i}\omega\mu}}\;\frac{M_{T}}{2\pi}\left( {\frac{M_{1}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{1}}}{r_{1}} - \frac{M_{2}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{2}}}{r_{2}}} \right)k}}} & \left( {22\; l} \right) \\{\mspace{79mu}{\frac{\mathbb{d}U_{t}}{\mathbb{d}k} = {{\mathbb{i}\omega\mu}\;\frac{M_{T}}{4\pi}\left( {{\frac{M_{1}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{1}}}{r_{1}}\left( {1 + {{\mathbb{i}}\;{kr}_{1}}} \right)} - {\frac{M_{2}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{2}}}{r_{2}}\left( {1 + {{\mathbb{i}}\;{kr}_{2}}} \right)}} \right)k}}} & \left( {22\; t} \right)\end{matrix}$

These close-form expressions no longer rely on the low-frequency, cubicpolynomial approximation. They serve to minimize the squared differencebetween measured induction-voltage data U_(meas) and their simulatedreproduction U_(simul).

$\begin{matrix}\begin{matrix}{L = {\frac{1}{2}\left( {U_{meas} - U_{simul}} \right)^{2}}} \\\left. \Rightarrow{\frac{\mathbb{d}}{\mathbb{d}k}L} \right. \\{= {\left( {U_{meas} - U_{simul}} \right)\frac{\mathbb{d}U_{simul}}{\mathbb{d}k}}} \\{= 0}\end{matrix} & (23)\end{matrix}$

The iteration uses a Newton-Raphson algorithm, assuming that the(n−1)^(st) iteration is only linearly different from the true value:

$\begin{matrix}{{U_{meas} - U_{simul}^{({n - 1})}} = {\frac{\mathbb{d}U_{simul}^{({n - 1})}}{\mathbb{d}k}\left( {k^{(n)} - k^{({n - 1})}} \right)}} & (24)\end{matrix}$

This relationship provides the iterative update for the wave numberk^((n)):k ^((n)) =k ^((n-1))+(U _(meas) −U _(simul) ^((n-1)))/(dU _(simul)^((n-1)) /dk)  (25)

The updated wave number is then used in the close-form expressions forresponses (21) and sensitivities (22) to determine the n^(th) iteration.

This iterative-inversion algorithm is the same for both polarizations;only the response (21) and sensitivity functions (22) differ. Thealgorithm converges rapidly, usually within three to four iterations,accurately reproducing both dielectric permittivity and electricconductivity. With the full electromagnetic wave e^(ikr) included theskin-effect correction is implicit in the algorithm.

For conductive media above 1000 mS/m (below 1 Ωm) the inversion for thetransversely polarized induction sonde tends to become unstable, leadingto negative permittivities that then diverge in the iteration. Here itproves necessary to impose a positivity constraint on both dielectricpermittivity and electric conductivity to ensure a good convergence.

In that regard, extending the iterative inversion algorithm from thelinear expression (24) described above to a higher-order polynomialexpression lead to improved performance with better numerical stabilityand more rapid convergence. Application of the principles herein to aquadratic and cubic expansion are provided below. Higher orderpolynomial expansions beyond cubic order are technically feasible butare not explicitly described at least because a person of skill couldgenerate such based on a review of this disclosure, and because anyadvantages of using higher-order polynomial expressions beyond cubicorder may be counter-balanced by the computational effort involved. Forpurposes of this specification, a higher order polynomial means aquadratic polynomial or greater. That is, for example, the phrase: “ahigher-order polynomial expression” excludes only a linear expression.

Equation (24) assumes that the measured induction signal U_(meas) islinearly approximated by the simulated signal U_(simul) and its firstderivative. The correaction is given as update in the wave number(k^((n))−k^((n-1))). A first generalization of equation (24) is anexpansion to quadratic order:

$\begin{matrix}{U_{meas} = {U_{simul}^{({n - 1})} + {\frac{\mathbb{d}U_{simul}^{({n - 1})}}{\mathbb{d}k}\left( {k^{(n)} - k^{({n - 1})}} \right)} + {\frac{1}{2}\frac{\mathbb{d}^{2}U_{simul}^{({n - 1})}}{\mathbb{d}k^{2}}\left( {k^{(n)} - k^{({n - 1})}} \right)^{2}}}} & (102)\end{matrix}$

A second generalization of equation (24) is the expansion to cubicorder:

$\begin{matrix}{U_{meas} = {U_{simul}^{({n - 1})} + {\frac{\mathbb{d}U_{simul}^{({n - 1})}}{\mathbb{d}k}\left( {k^{(n)} - k^{({n - 1})}} \right)} + {\frac{1}{2}\frac{\mathbb{d}^{2}U_{simul}^{({n - 1})}}{\mathbb{d}k^{2}}\left( {k^{(n)} - k^{({n - 1})}} \right)^{2}} + {\frac{1}{6}\frac{\mathbb{d}^{3}U_{simul}^{({n - 1})}}{\mathbb{d}k^{3}}\left( {k^{(n)} - k^{({n - 1})}} \right)^{3}}}} & (103)\end{matrix}$

As with the linear algorithm, both the quadratic and cubic algorithmsprovide an iterative update for the wavenumber k^((n)). With respect toapplication of the iterative update to the quadratic order, equation(102) is first re-written as quadratic equation in the wave-numberupdate (k^((n))−k^((n-1))).

$\begin{matrix}{{\left( {k^{(n)} - k^{({n - 1})}} \right)^{2} + {2\frac{{\mathbb{d}U_{simul}^{({n - 1})}}/{\mathbb{d}k}}{{\mathbb{d}^{2}U_{simul}^{({n - 1})}}/{\mathbb{d}k^{2}}}\left( {k^{(n)} - k^{({n - 1})}} \right)} + {2\frac{U_{simul}^{({n - 1})} - U_{meas}}{{\mathbb{d}^{2}U_{simul}^{({n - 1})}}/{\mathbb{d}k^{2}}}}} = 0} & (104)\end{matrix}$

Equation (104) can then be used to solve for the update:

$\begin{matrix}{k^{(n)} = {k^{({n - 1})} - {\frac{{\mathbb{d}U_{simul}^{({n - 1})}}/{\mathbb{d}k}}{{\mathbb{d}^{2}U_{simul}^{({n - 1})}}/{\mathbb{d}k^{2}}} \pm \sqrt{\left( \frac{{\mathbb{d}U_{simul}^{({n - 1})}}/{\mathbb{d}k}}{{\mathbb{d}^{2}U_{simul}^{({n - 1})}}/{\mathbb{d}k^{2}}} \right)^{2} - {2\frac{U_{simul}^{({n - 1})} - U_{meas}}{{\mathbb{d}^{2}U_{simul}^{({n - 1})}}/{\mathbb{d}k^{2}}}}}}}} & (105)\end{matrix}$

Equation (105) replaces the linear update equation (25) when applyingthe quadratic iterative inversion algorithm.

It is noted that equation (105) gives two solutions; however, only thepositive sign is the physically correct solution, sinceU_(simul)=U_(meas) would render any further iteration unnecessary; hencethe leading term and the square root must cancel each other.

The only additional calculations needed with respect to equation (105)beyond those provided for in connection with applying the lineariterative algorithm, are calculations of the second derivatives inequations (102, 104, 105) from the first derivatives in equations (22):

$\begin{matrix}{\frac{\mathbb{d}^{2}U_{l}}{\mathbb{d}k^{2}} = {{\frac{\mathbb{d}}{\mathbb{d}k}\frac{\mathbb{d}U_{l}}{\mathbb{d}k}} = {{- {\mathbb{i}\omega\mu}}\;{\frac{M_{T}}{2\pi}\left\lbrack {\left( {\frac{M_{1}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{1}}}{r_{1}} - \frac{M_{2}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{2}}}{r_{2}}} \right) + {{\mathbb{i}}\;{k\left( {{M_{1}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{1}}} - {M_{2}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{2}}}} \right)}}} \right\rbrack}}}} & \left( {106\; l} \right) \\{\frac{\mathbb{d}^{2}U_{t}}{\mathbb{d}k^{2}} = {{\frac{\mathbb{d}}{\mathbb{d}k}\frac{\mathbb{d}U_{t}}{\mathbb{d}k}} = {{\mathbb{i}\omega\mu}\;{\frac{M_{T}}{4\pi}\begin{bmatrix}{\left( {\frac{M_{1}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{1}}}{r_{1}} - \frac{M_{2}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{2}}}{r_{2}}} \right) + {3{\mathbb{i}}\;{k\left( {{M_{1}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{1}}} - {M_{2}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{2}}}} \right)}} -} \\{- {k^{2}\left( {{M_{1}r_{1}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{1}}} - {M_{2}r_{2}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{2}}}} \right)}}\end{bmatrix}}}}} & \left( {106\; t} \right)\end{matrix}$

The updated wave number k^((n)) in equation (105) is now used to computeand update U_(simul) ^((n)) in equations (21).

With respect to application of the iterative update to the cubic order,equation (103) is first re-written as cubic equation in the wave-numberupdate (k^((n))−k^(n-1))):

$\begin{matrix}{{\left( {k^{(n)} - k^{({n - 1})}} \right)^{3} + {3\frac{{\mathbb{d}^{2}U_{simul}^{({n - 1})}}/{\mathbb{d}k^{2}}}{{\mathbb{d}^{3}U_{simul}^{({n - 1})}}/{\mathbb{d}k^{3}}}\left( {k^{(n)} - k^{({n - 1})}} \right)^{2}} + {6\frac{{\mathbb{d}U_{simul}^{({n - 1})}}/{\mathbb{d}k}}{{\mathbb{d}^{3}U_{simul}^{({n - 1})}}/{\mathbb{d}k^{3}}}\left( {k^{(n)} - k^{({n - 1})}} \right)} + {6\frac{U_{simul}^{({n - 1})} - U_{meas}}{{\mathbb{d}^{3}U_{simul}^{({n - 1})}}/{\mathbb{d}k^{3}}}}} = 0} & (107)\end{matrix}$

This cubic polynomial is solved in closed form as described above inconnection with equations 16-20 (including equations 20a, 20b and 20c).In the present context of a cubic iterative inversion algorithm, thecoefficients d and e in equation (18) are:

$\begin{matrix}{d = {{\frac{c_{1}}{3} - \frac{c_{2}^{2}}{9}} = {{2\frac{{\mathbb{d}U_{simul}^{({n - 1})}}/{\mathbb{d}k}}{{\mathbb{d}^{3}U_{simul}^{({n - 1})}}/{\mathbb{d}k^{3}}}} - {\frac{1}{3}\left( \frac{{\mathbb{d}^{2}U_{simul}^{({n - 1})}}/{\mathbb{d}k^{2}}}{{\mathbb{d}^{3}U_{simul}^{({n - 1})}}/{\mathbb{d}k^{3}}} \right)^{2}}}}} & (108) \\\begin{matrix}{e = {{\frac{1}{6}\left( {{c_{1}c_{2}} - {3c_{0}}} \right)} - \frac{c_{2}^{3}}{27}}} \\{= {{3\frac{{\mathbb{d}^{2}U_{simul}^{({n - 1})}}/{\mathbb{d}k^{2}}}{{\mathbb{d}^{3}U_{simul}^{({n - 1})}}/{\mathbb{d}k^{3}}}\frac{{\mathbb{d}U_{simul}^{({n - 1})}}/{\mathbb{d}k}}{{\mathbb{d}^{3}U_{simul}^{({n - 1})}}/{\mathbb{d}k^{3}}}} -}} \\{{3\frac{U_{simul}^{({n - 1})} - U_{meas}}{{\mathbb{d}^{3}U_{simul}^{({n - 1})}}/{\mathbb{d}k^{3}}}} - {\frac{2}{9}\left( \frac{{\mathbb{d}^{2}U_{simul}^{({n - 1})}}/{\mathbb{d}k^{2}}}{{\mathbb{d}^{3}U_{simul}^{({n - 1})}}/{\mathbb{d}k^{3}}} \right)^{3}}}\end{matrix} & (109)\end{matrix}$

The coefficients f₁ and f₂ (see equation 19 above) and the finalsolutions z₁, z₂, z₃ (see equations 20 above) all constitute lengthyexpressions and are readily coded in a computer program. Only onesolution in equations (20) will provide physically reasonable results;the other two solutions lead to negative conductivities orpermittivities, which is not a physical reality in the present context.

Finally, the third derivatives d³U_(l,t)/dk³ as derivatives of theexpressions (106) should be calculated:

$\begin{matrix}{\frac{\mathbb{d}^{3}U_{l}}{\mathbb{d}k^{3}} = {{\frac{\mathbb{d}}{\mathbb{d}k}\frac{\mathbb{d}^{2}U_{l}}{\mathbb{d}k^{2}}} = {{- {\mathbb{i}\omega\mu}}\;{\frac{M_{T}}{2\pi}\left\lbrack {{2{{\mathbb{i}}\left( {{M_{1}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{1}}} - {M_{2}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{2}}}} \right)}} - {k\left( {{M_{1}r_{1}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{1}}} - {M_{2}r_{2}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{2}}}} \right)}} \right\rbrack}}}} & \left( {110\; l} \right) \\{\frac{\mathbb{d}^{3}U_{t}}{\mathbb{d}k^{3}} = {{\frac{\mathbb{d}}{\mathbb{d}k}\frac{\mathbb{d}^{2}U_{t}}{\mathbb{d}k^{2}}} = {{\mathbb{i}\omega\mu}\;{\frac{M_{T}}{4\pi}\begin{bmatrix}{{4{i\left( {{M_{1}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{1}}} - {M_{2}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{2}}}} \right)}} - {5{k\left( {{M_{1}r_{1}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{1}}} - {M_{2}r_{2}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{2}}}} \right)}} -} \\{{- {\mathbb{i}}}\;{k^{2}\left( \left( {{M_{1}r_{1}^{2}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{1}}} - {M_{2}r_{2}^{2}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{2}}}} \right) \right)}}\end{bmatrix}}}}} & \left( {110\; t} \right)\end{matrix}$

The present invention relies on the measurement of the quadrature signalwith a comparable accuracy and precision as the in-phase-signalmeasurement (as is possible with current induction tools). The wavenumber is considered in its complete form (3), including the dielectricpermittivity; the induction approximation of negligible permittivity isabandoned.

$\begin{matrix}{k = {\frac{\omega}{c}\sqrt{\mu_{r}}\sqrt{ɛ_{r} + {{\mathbb{i}}\frac{\sigma}{{\omega ɛ}_{0}}}}}} & (3)\end{matrix}$Only the relative magnetic permeability is assumed to be μ_(r)=1.

The low-frequency expansion to cubic order in the wave number k(equation 9) can be solved in closed form: the first root of the cubicpolynomial constitutes an improved solution for the wave number andhence simultaneously for the apparent dielectric permittivity andelectric conductivity. This cubic polynomial inversion implicitlycontains the skin-effect correction.

The cubic polynomial inversion can provide an improved apparent electricconductivity, while the apparent dielectric permittivity can tend to beof poor quality. An iterative inversion algorithm provides accurate,reliable and numerically stable results for both dielectric permittivityand electric conductivity from R- and X-signals simultaneously.

This iterative inversion algorithm implicitly uses the fact thatpermittivity and conductivity are combined into a single, complex-valuedquantity and that the R- and X-signal combined as complex-valuedquantity constitute a holomorphic function of the complex permittivity.The holomorphic character of the signal renders the Cauchy-Riemannidentities valid in the signal sensitivity with respect to the wavenumber.

It will be appreciated that various changes can be made while stayingwithin the scope of the invention.

The invention claimed is:
 1. A method of inverting induction loggingdata for evaluating the properties of underground formations surroundinga borehole, the data comprising induction voltage measurements obtainedfrom a tool placed close to the formations of interest, the methodcomprising: (a) operating a tool in a borehole to obtain inductionvoltage data; (b) defining a relationship relating the induction voltageto wave number, dielectric permittivity and conductivity; defining acubic polynomial expansion of the relationship; and solving the cubicpolynomial relationship using the voltage measurements to obtain valuesfor conductivity that includes skin-effect correction and apparentdielectric permittivity; and (c) using the obtained values forconductivity and apparent dielectric permittivity to derive a simulatedvalue of induction voltage; determining the difference between thesimulated value of the induction voltage and the measured inductionvoltage; and using a higher-order polynomial expression to iterativelyupdate the values of conductivity and dielectric permittivity used forthe derivation of the simulated value of induction voltage to minimiseits difference with respect to the measured value.
 2. A method asclaimed in claim 1, wherein the relationship relating induction voltageto wave number, dielectric permittivity and conductivity is:$k = {\frac{\omega}{c}\sqrt{\mu_{r}}\sqrt{ɛ_{r} + {{\mathbb{i}}\frac{\sigma}{{\omega ɛ}_{0}}}}}$where k is wave number, ω is circular frequency, c is the speed of lightin a vacuum, μ_(r) is relative magnetic permeability, ∈_(r) is relativepermittivity, ∈₀ is the absolute dielectric permittivity of a vacuum andσ is conductivity.
 3. A method as claimed in claim 1, further comprisingderiving roots of the cubic polynomial and using at least one of theroots to obtain the wave number.
 4. A method as claimed in claim 3,further comprising ignoring roots giving physically impossible valuesfor parameters of interest.
 5. A method as claimed in claim 1, whereinthe step of using a higher-order polynomial expression to iterativelyupdate the values of conductivity and dielectric permittivity used forthe derivation of the simulated value of induction voltage to minimiseits difference with respect to the measured value, comprises minimisinga squared difference between measured induction voltage U_(meas) and asimulated reproduction U_(simul) according to the relationship:$L = {\left. {\frac{1}{2}\left( {U_{meas} - U_{simul}} \right)^{2}}\Rightarrow{\frac{\mathbb{d}}{\mathbb{d}k}L} \right. = {{\left( {U_{meas} - U_{simul}} \right)\frac{\mathbb{d}U_{simul}}{\mathbb{d}k}} = 0.}}$6. A method as claimed in claim 5, wherein the higher order polynomialexpression is a quadratic expression according to:$U_{meas} = {U_{simul}^{({n - 1})} + {\frac{\mathbb{d}U_{simul}^{({n - 1})}}{\mathbb{d}k}\left( {k^{(n)} - k^{({n - 1})}} \right)} + {\frac{1}{2}\frac{\mathbb{d}^{2}U_{simul}^{({n - 1})}}{\mathbb{d}k^{2}}\left( {k^{(n)} - k^{({n - 1})}} \right)^{2}}}$and the wave number k^((n)) is updated according to the relationship:$\begin{matrix}{k^{(n)} = {k^{({n - 1})} - \frac{{\mathbb{d}U_{simul}^{({n - 1})}}/{\mathbb{d}k}}{{\mathbb{d}^{2}U_{simul}^{({n - 1})}}/{\mathbb{d}k^{2}}}}} \\{\pm {\sqrt{\left( \frac{{\mathbb{d}U_{simul}^{({n - 1})}}/{\mathbb{d}k}}{{\mathbb{d}^{2}U_{simul}^{({n - 1})}}/{\mathbb{d}k^{2}}} \right)^{2} - {2\frac{U_{simul}^{({n - 1})} - U_{meas}}{{\mathbb{d}^{2}U_{simul}^{({n - 1})}}/{\mathbb{d}k^{2}}}}}.}}\end{matrix}$
 7. A method as claimed in claim 6, further comprisingderiving roots of the cubic polynomial and ignoring roots givingphysically impossible values for parameters of interest.
 8. A method asclaimed in claim 5, wherein the higher order polynomial expression is acubic expression according to:$U_{meas} = {U_{simul}^{({n - 1})} + {\frac{\mathbb{d}U_{simul}^{({n - 1})}}{\mathbb{d}k}\left( {k^{(n)} - k^{({n - 1})}} \right)} + {\frac{1}{2}\frac{\mathbb{d}^{2}U_{simul}^{({n - 1})}}{\mathbb{d}k^{2}}\left( {k^{(n)} - k^{({n - 1})}} \right)^{2}} + {\frac{1}{6}\frac{\mathbb{d}^{3}U_{simul}^{({n - 1})}}{\mathbb{d}k^{3}}\left( {k^{(n)} - k^{({n - 1})}} \right)^{3}}}$and the wave number k^((n)) is updated according to the relationship:${\left( {k^{(n)} - k^{({n - 1})}} \right)^{3} + {3\frac{{\mathbb{d}^{2}U_{simul}^{({n - 1})}}/{\mathbb{d}k^{2}}}{{\mathbb{d}^{3}U_{simul}^{({n - 1})}}/{\mathbb{d}k^{3}}}\left( {k^{(n)} - k^{({n - 1})}} \right)^{2}} + {6\frac{{\mathbb{d}U_{simul}^{({n - 1})}}/{\mathbb{d}k}}{{\mathbb{d}^{3}U_{simul}^{({n - 1})}}/{\mathbb{d}k^{3}}}\left( {k^{(n)} - k^{({n - 1})}} \right)} + {6\frac{U_{simul}^{({n - 1})} - U_{meas}}{{\mathbb{d}^{3}U_{simul}^{({n - 1})}}/{\mathbb{d}k^{3}}}}} = 0.$9. A method as claimed in claim 8, further comprising deriving roots ofthe cubic polynomial and ignoring roots giving physically impossiblevalues for parameters of interest.
 10. A method as claimed in claim 1,wherein the induction voltage measurements comprise in-phase andquadrature measurements of substantially the same accuracy andresolution.
 11. A method as claimed in claim 1, wherein the inductionlogging data is obtained from a three-coil tool comprising a transmittercoil with magnetic dipole moment M_(T), a main receiver coil withmagnetic dipole moment M₁ at distance r₁ from the transmitter coil, anda bucking coil with a magnetic moment M₂ at distance r₂ from thetransmitter coil, the dipole moments being aligned substantiallyparallel to the axis of the tool, the cubic polynomial expansioncomprising:$U_{l} = {{- {\mathbb{i}\omega\mu}}\;\frac{M_{T}}{2\pi}\left( {{\left( {\frac{M_{1}}{2r_{1}} - \frac{M_{2}}{2r_{2}}} \right)k^{2}} + {{{\mathbb{i}}\left( {\frac{M_{1}}{3} - \frac{M_{2}}{3}} \right)}k^{3}}} \right)}$wherein U_(l) is the induction voltage.
 12. A method as claimed in claim11, wherein the simulated value of induction voltage is derivedaccording to the relationship$U_{l} = {{- {\mathbb{i}\omega\mu}}\;\frac{M_{T}}{2\pi}\left( {{\frac{M_{1}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{1}}}{r_{1}^{3}}\left( {1 - {{\mathbb{i}}\;{kr}_{1}}} \right)} - {\frac{M_{2}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{2}}}{r_{2}^{3}}\left( {1 - {{\mathbb{i}}\;{kr}_{2}}} \right)}} \right)}$using a sensitivity function derived according to the relationship:$\frac{\mathbb{d}U_{l}}{\mathbb{d}k} = {{- {\mathbb{i}\omega\mu}}\;\frac{M_{T}}{2\pi}\left( {\frac{M_{1}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{1}}}{r_{1}} - \frac{M_{2}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{2}}}{r_{2}}} \right)k}$where e^(ikr) is the full electromagnetic wave.
 13. A method as claimedin claim 1, wherein the induction logging data is obtained from athree-coil tool comprising a transmitter coil with magnetic dipolemoment M_(T), a main receiver coil with magnetic dipole moment M₁ atdistance r₁ from the transmitter coil, and a bucking coil with amagnetic moment M₂ at distance r₂ from the transmitter coil, the dipolemoments being aligned substantially transverse to the axis of the tool,the cubic polynomial expansion comprising:$U_{t} = {{- {\mathbb{i}\omega\mu}}\;\frac{M_{T}}{4\pi}\left( {{\left( {\frac{M_{1}}{2r_{1}} - \frac{M_{2}}{2r_{2}}} \right)k^{2}} + {2{{\mathbb{i}}\left( {\frac{M_{1}}{3} - \frac{M_{2}}{3}} \right)}k^{3}}} \right)}$wherein U_(t) is the induction voltage.
 14. A method as claimed in claim13, wherein the simulated value of induction voltage is derivedaccording to the relationship$U_{t} = {{- {\mathbb{i}\omega\mu}}\;\frac{M_{T}}{4\pi}\left( {{\frac{M_{1}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{1}}}{r_{1}^{3}}\left( {1 - {{\mathbb{i}}\;{kr}_{1}} - {k^{2}r_{1}^{2}}} \right)} - {\frac{M_{2}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{2}}}{r_{2}^{3}}\left( {1 - {{\mathbb{i}}\;{kr}_{2}} - {k^{2}r_{2}^{2}}} \right)}} \right)}$using a sensitivity function derived according to the relationship:$\frac{\mathbb{d}U_{t}}{\mathbb{d}k} = {{\mathbb{i}\omega\mu}\;\frac{M_{T}}{4\pi}\left( {{\frac{M_{1}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{1}}}{r_{1}}\left( {1 + {{\mathbb{i}}\;{kr}_{1}}} \right)} - {\frac{M_{2}{\mathbb{e}}^{{\mathbb{i}}\;{kr}_{2}}}{r_{2}}\left( {1 + {{\mathbb{i}}\;{kr}_{2}}} \right)}} \right)k}$where e^(ikr) is the full electromagnetic wave.
 15. A method ofinverting induction logging data for evaluating the properties ofunderground formations surrounding a borehole, the data comprisinginduction voltage measurements obtained from a tool placed close to theformations of interest, the method comprising: (a) operating a tool in aborehole to obtain induction voltage data; (b) defining a relationshiprelating the induction voltage to wave number, dielectric permittivityand conductivity; defining a n>2 polynomial expansion of therelationship, wherein the n>2 polynomial expansion comprises a cubicpolynomial expansion, a higher-order polynomial expansion, orcombinations thereof; and solving the n>2 polynomial relationship usingthe voltage measurements to obtain values for conductivity that includesskin-effect correction and apparent dielectric permittivity; and (c)using the obtained values for conductivity and apparent dielectricpermittivity to derive a simulated value of induction voltage;determining the difference between the simulated value of the inductionvoltage and the measured induction voltage; and using a n+1 polynomialexpression to iteratively update the values of conductivity anddielectric permittivity used for the derivation of the simulated valueof induction voltage to minimise its difference with respect to themeasured value.
 16. A method as claimed in claim 15, wherein therelationship relating induction voltage to wave number, dielectricpermittivity and conductivity is:$k = {\frac{\omega}{c}\sqrt{\mu_{r}}\sqrt{ɛ_{r} + {{\mathbb{i}}\frac{\sigma}{{\omega ɛ}_{0}}}}}$where k is wave number, ω is circular frequency, c is the speed of lightin a vacuum, μ_(r) is relative magnetic permeability, ∈_(r) is relativepermittivity, ∈₀ is the absolute dielectric permittivity of a vacuum andσ is conductivity.
 17. A method as claimed in claim 15, furthercomprising deriving roots of the n>2 polynomial and using at least oneof the roots to obtain the wave number.
 18. A method as claimed in claim17, further comprising ignoring roots giving physically impossiblevalues for parameters of interest.
 19. A method as claimed in claim 15,wherein the induction voltage measurements comprise in-phase andquadrature measurements of substantially the same accuracy andresolution.
 20. A method as claimed in claim 15, wherein the inductionlogging data is obtained from a three-coil tool comprising a transmittercoil with magnetic dipole moment M_(T), a main receiver coil withmagnetic dipole moment M₁ at distance r₁ from the transmitter coil, anda bucking coil with a magnetic moment M₂ at distance r₂ from thetransmitter coil, the dipole moments being aligned substantiallyparallel to the axis of the tool, the cubic polynomial expansioncomprising:$U_{l} = {{- {\mathbb{i}\omega\mu}}\;\frac{M_{T}}{2\pi}\left( {{\left( {\frac{M_{1}}{2r_{1}} - \frac{M_{2}}{2r_{2}}} \right)k^{2}} + {{{\mathbb{i}}\left( {\frac{M_{1}}{3} - \frac{M_{2}}{3}} \right)}k^{3}}} \right)}$wherein U_(l) is the induction voltage.